Table Of Content

Cover page Page: i Halftitle page Page: i Series page Page: ii Title page Page: xi Copyright page Page: xii Contents Page: xiii List of illustrations Page: xv Chapter 1 What is number theory? Page: xvi Integers Page: 4 Squares and cubes Page: 6 Perfect numbers Page: 9 Prime numbers Page: 10 Chapter 2 Multiplying and dividing Page: 13 Multiples and divisors Page: 14 Least common multiple and greatest common divisor Page: 17 Euclid’s algorithm Page: 22 Squares Page: 26 Divisor tests Page: 30 Chapter 3 Prime-time mathematics Page: 37 The sieve of Eratosthenes Page: 38 Primes go on for ever Page: 41 Factorizing into primes Page: 42 Searching for primes Page: 45 Chapter 4 Congruences, clocks, and calendars Page: 57 Clock arithmetic Page: 58 Congruences and the calendar Page: 63 Solving linear congruences Page: 66 Chapter 5 More triangles and squares Page: 78 Linear Diophantine equations Page: 79 Right-angled triangles Page: 82 Sums of squares Page: 85 Higher powers Page: 90 Chapter 6 From cards to cryptography Page: 96 Fermat’s little theorem Page: 97 Generalizing Fermat’s little theorem Page: 102 Factorizing large numbers Page: 108 Chapter 7 Conjectures and theorems Page: 111 Two famous conjectures Page: 112 The distribution of primes Page: 116 Primes in arithmetic progressions Page: 122 Unique factorization Page: 126 Chapter 8 How to win a million dollars Page: 129 Infinite series Page: 132 The zeta function Page: 134 The Riemann hypothesis Page: 137 Consequences Page: 140 Chapter 9 Aftermath Page: 141 The first ten questions Page: 142 Integers Page: 144 Squares and cubes Page: 145 Perfect numbers Page: 147 Prime numbers Page: 147 Further reading Page: 151 Index Page: 153 Economics Page: 157 Information Page: 158 Innovation Page: 159 Nothing Page: 160

Description:

Number theory is the branch of mathematics that is primarily concerned with the counting numbers. Of particular importance are the prime numbers, the 'building blocks' of our number system. The subject is an old one, dating back over two millennia to the ancient Greeks, and for many years has been studied for its intrinsic beauty and elegance, not least because several of its challenges are so easy to state that everyone can understand them, and yet no-one has ever been able to resolve them. But number theory has also recently become of great practical importance - in the area of cryptography, where the security of your credit card, and indeed of the nation's defence, depends on a result concerning prime numbers that dates back to the 18th century. Recent years have witnessed other spectacular developments, such as Andrew Wiles's proof of 'Fermat's last theorem' (unproved for over 250 years) and some exciting work on prime numbers. In this Very Short Introduction Robin Wilson introduces the main areas of classical number theory, both ancient and modern. Drawing on the work of many of the greatest mathematicians of the past, such as Euclid, Fermat, Euler, and Gauss, he situates some of the most interesting and creative problems in the area in their historical context. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.